In the previous example, we performed the decomposition of the roots (and the ensuing decomposition of the algebra) with respect to one of the simple roots which then defined the level. In general, one may consider a similar decomposition of the roots of a rank Kac–Moody algebra with respect to an arbitrary number of the simple roots and then the level is generalized to the “multilevel” .
We consider a Kac–Moody algebra of rank and we let be a finite regular rank subalgebra of whose Dynkin diagram is obtained by deleting a set of nodes from the Dynkin diagram of .
Let be a root of ,gradation of and which can be written formally as
We note for further reference that the following structure is inherited from the gradation:the subspaces at definite values of the multilevel are invariant subspaces under the adjoint action of . In other words, the action of on does not change the coefficients .
At level zero, , the representation of the subalgebra in the subspace contains the adjoint representation of , just as in the case of discussed in Section 8.1. All positive and negative roots of are relevant. Level zero contains in addition singlets for each of the Cartan generator associated to the set .
Whenever one of the ’s is positive, all the other ones must be non-negative for the subspace to be nontrivial and only positive roots appear at that value of the multilevel.
Let be the module of a representation of and be one of the weights occurring in the representation. We define the action of in the representation on as). Any representation of is also a representation of . When restricted to the Cartan subalgebra of , defines a weight , which one can realize geometrically as follows.
The dual space may be viewed as the -dimensional subspace of spanned by the simple roots , . The metric induced on that subspace is positive definite since is finite-dimensional. This implies, since we assume that the metric on is nondegenerate, that can be decomposed as the direct sum
It follows that one can identify the weight with the orthogonal projection of on . This is true, in particular, for the fundamental weights . The fundamental weights project on for and project on the fundamental weights of the subalgebra for . These are also denoted . For a general weight, one has
There is an interesting relationship between root multiplicities in the Kac–Moody algebra and weight multiplicites of the corresponding -weights, which we will explore here.
For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real roots of indefinite Kac–Moody algebras. However, as pointed out in Section 4, the imaginary roots can have arbitrarily large multiplicity. This must therefore be taken into account in the sum (8.13).
Let be a root of . There are two important ingredients:
It follows that the root multiplicity of is given as a sum over its multiplicities as a weight in the various representations at level . Some representations can appear more than once at each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outer multiplicity , which counts the number of times each representation appears at level . So, for each representation at level we must count the individual weight multiplicities in that representation and also the number of times this representation occurs. The total multiplicity of can then be written as
The subspaces can now be written explicitly as
Finally, we mention that the multiplicity of a root can be computed recursively using the Peterson recursion relation, defined as , we call this the co-multiplicity. Note that if is not a root, this gives no contribution to the co-multiplicity. Another feature of the co-multiplicity is that even if the multiplicity of some root is zero, the associated co-multiplicity does not necessarily vanish. Taking advantage of the fact that all real roots have multiplicity one it is possible, in principle, to compute recursively the multiplicity of any imaginary root. Since no closed formula exists for the outer multiplicity , one must take a detour via the Peterson relation and Equation (8.25) in order to find the outer multiplicity of each representation at a given level. We give in Table 38 a list of root multiplicities and co-multiplicities of roots of up to height 10.
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